Towards improved analysis of short mesoscale sea level signals from satellite altimetry
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Earth Syst. Sci. Data, 14, 1493–1512, 2022
https://doi.org/10.5194/essd-14-1493-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.
Towards improved analysis of short mesoscale
sea level signals from satellite altimetry
Yves Quilfen, Jean-François Piolle, and Bertrand Chapron
Laboratoire d’Océanographie Physique et Spatiale (LOPS), IFREMER,
Univ. Brest, CNRS, IRD, IUEM, Brest, France
Correspondence: Yves Quilfen (yquilfen@ifremer.fr)
Received: 18 October 2021 – Discussion started: 26 October 2021
Revised: 22 February 2022 – Accepted: 24 February 2022 – Published: 4 April 2022
Abstract. Satellite altimeters routinely supply sea surface height (SSH) measurements, which are key observa-
tions for monitoring ocean dynamics. However, below a wavelength of about 70 km, along-track altimeter mea-
surements are often characterized by a dramatic drop in signal-to-noise ratio (SNR), making it very challenging
to fully exploit the available altimeter observations to precisely analyze small mesoscale variations in SSH. Al-
though various approaches have been proposed and applied to identify and filter noise from measurements, no
distinct methodology has emerged for systematic application in operational products. To best address this un-
resolved issue, the Copernicus Marine Environment Monitoring Service (CMEMS) actually provides simple
band-pass filtered data to mitigate noise contamination of along-track SSH signals. More innovative and suitable
noise filtering methods are thus left to users seeking to unveil small-scale altimeter signals. As demonstrated
here, a fully data-driven approach is developed and applied successfully to provide robust estimates of noise-free
sea level anomaly (SLA) signals (Quilfen, 2021). The method combines empirical mode decomposition (EMD),
used to help analyze non-stationary and non-linear processes, and an adaptive noise filtering technique inspired
by discrete wavelet transform (DWT) decompositions. It is found to best resolve the distribution of SLA vari-
ability in the 30–120 km mesoscale wavelength band. A practical uncertainty variable is attached to the denoised
SLA estimates that accounts for errors related to the local SNR but also for uncertainties in the denoising pro-
cess, which assumes that the SLA variability results in part from a stochastic process. For the available period,
measurements from the Jason-3, Sentinel-3, and SARAL/AltiKa missions are processed and analyzed, and their
energy spectral and seasonal distributions are characterized in the small mesoscale domain. In anticipation of the
upcoming SWOT (Surface Water and Ocean Topography) mission data, the SASSA (Satellite Altimeter Short-
scale Signals Analysis, https://doi.org/10.12770/1126742b-a5da-4fe2-b687-e64d585e138c, Quilfen and Piolle,
2021) data set of denoised SLA measurements for three reference altimeter missions has already been shown to
yield valuable opportunities to evaluate global small mesoscale kinetic energy distributions.
1 Introduction range (e.g., Fu, 1983; Le Traon et al., 2008; Dufau et al.,
2016; Tchibilou et al., 2018) or the role of upper-ocean sub-
Satellite altimetry has supported studies related to ocean dy- mesoscale dynamics that is critical to the transport of heat be-
namics for more than 25 years, often looking to push the tween the ocean interior and the atmosphere (Su et al., 2018).
limits of these observations to capture ocean motions at ever One of the main limitations is that altimetry measure-
smaller scales. New paradigms are thus emerging from this ments are often characterized by a low signal-to-noise ratio
observational effort, among them the distinction between (SNR), which has a significant impact on geophysical analy-
balanced and unbalanced motions that can lead to charac- sis capability at spatial scales smaller than 120 km. The main
teristic changes in sea surface height (SSH) signal varia- sources of noise are induced by instrumental white noise,
tions and associated spectrum in the 30–200 km wavelength
Published by Copernicus Publications.
1494 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals errors related to processing, including the retracking algo- sian noise distribution is predictable (Flandrin et al., 2004). rithm and corrections, and errors related to the intrinsic vari- Noise removal strategies can then be developed with re- ability of radar echoes in the altimeter footprint that causes sults often superior to wavelet-based techniques (Kopsi- the notorious spectral hump in 20 and 1 Hz data (Sandwell nis and McLaughin, 2009). An EMD-based technique was and Smith, 2005; Dibarboure et al., 2014). Furthermore, be- successfully applied to altimetry data to more precisely cause retrieved parameters are obtained from the same wave- analyze along-track altimeter SWH measurements to map form retracking algorithm, they have highly correlated er- wave–current interactions (Quilfen et al., 2018; Quilfen and rors, i.e., the standard MLE4 processing produces four es- Chapron, 2019) known to predominate at scales smaller than timated parameters with correlated errors (SSH; significant 100 km. In particular, the method is suitable for processing wave height, SWH; sigma-0; and off-nadir angle). These er- non-stationary and non-linear signals, and thus for accurate rors directly limit the accuracy of the SSH measurement, re- and consistent recovery of strong gradients and extreme val- quiring advanced denoising techniques (Quartly et al., 2021). ues. Building on local noise analysis, the denoising of small Analysis of fine-scale ocean dynamics therefore requires mesoscale signals is performed on an adaptive basis to the preliminary noise filtering, and low-pass or smoothing filters local SNR. A detailed description of the EMD denoising ap- (e.g., Lanczos, running mean, or Loess filter) are frequently proach applied to satellite altimetry data is given in Quilfen used. These filters effectively smoothen altimeter signals, and Chapron (2021). but result in the systematic loss of small-scale (< ∼ 70 km) In this paper, the method is extended to more thoroughly geophysical information, only remove high-frequency (HF) evaluate an experimental data set of denoised sea level noise, and can produce artifacts in the analyzed geophys- anomaly (SLA) measurements, from three reference altime- ical variability. Considering that a major source of high- ters, the Jason-3, Sentinel-3, and SARAL/AltiKa, in order frequency SSH errors is associated with the correlation be- to capture short mesoscale information. Section 2 provides tween SSH and SWH errors through the retracking algo- a description of the data sets used and Sect. 3 describes rithm and sea state bias correction, recent approaches pro- the denoising methodology main principles. In Sect. 4, pose deriving a statistical correction to mitigate correlated which presents the results, examples of denoised SLA sig- high-frequency SSH errors (Zaron and De Carvalho, 2016; nals are given, and the energy spectral and seasonal distri- Quartly, 2019; Tran et al., 2021). Both the spectral hump butions of denoised measurements are characterized in the and white noise variance at 20 Hz are indeed significantly small mesoscale domain for these three altimeters. Section 5 reduced. Yet, these approaches are based on the assump- presents key features for comparison with other distributed tion that a separation scale between SWH noise and SWH data sets that make our approach more attractive. A discus- geophysical information can be defined in order to apply a sion follows, analyzing the main results, and a summary is low-pass filter on SWH along-track signals and to estimate given. Appendices A and B provide details on the denois- the correlated SSH and SWH errors at high frequency. In ing scheme and power spectral density (PSD) calculation, practice, a cut-off wavelength of 140 km is used in Tran et respectively. al. (2021). However, this approach has a fundamental caveat, as wave–current interactions strongly impact surface waves and current dynamics at short mesoscale and sub-mesoscale 2 Data wavelengths (e.g., Kudryavtsev et al., 2017; Ardhuin et al., 2017; McWilliams, 2018; Quilfen et al., 2018; Quilfen and The Copernicus Marine Environment Service (CMEMS) Chapron, 2019; Romero et al., 2020; Villas Bôas et al., is responsible for the dissemination of various satel- 2020). Wave–current interactions are ubiquitous phenomena, lite altimeter products, among which the level 3 along- and current-induced SWH variability at scales smaller than track SSHs distributed in delayed mode (product identifier: 100 km can be expected depending upon the strength of the SEALEVEL_GLO_PHY_L3_REP_OBSERVATIONS_008 current gradient relative to the wavelength and direction of _062) are the state-of-the-art product that takes into account propagation of surface waves. Applying a correction based the various improvements proposed in the framework of the on the assumption that SWH variability is primarily all noise SSALTO/DUACS activities (Taburet et al., 2021). The input below 100 km will therefore likely affect small mesoscale data quality control verifies that the system uses the best al- SSH signals in various and complex ways. timeter data. From these products, which include data from To overcome these difficulties, an adaptive noise removal all altimetry missions, we use the “unfiltered SLA” variable approach for satellite altimeter measurements has been de- to derive our analysis of SLA measurements. rived. It is based on the non-parametric empirical mode The present study aims to provide research products, the decomposition (EMD) method developed to analyze non- SASSA (Satellite Altimeter Short-scale Signals Analysis) stationary and non-linear signals (Huang et al., 1998; Huang data set, and innovative solutions for better exploitation of and Wu, 2008). EMD is a scale decomposition of a discrete the mesoscale mapping capabilities of altimeters. The anal- signal into a limited number of amplitude- and frequency- ysis is therefore limited to three current altimeter missions, modulated functions (AM/FM), among which the Gaus- Jason-3, Sentinel-3, and SARAL/AltiKa, each carrying an Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022
Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1495
instrument with particular distinctive characteristics. The Huang and Wu, 2008) and its filter bank characteristics when
Jason-3 altimeter is the reference dual-frequency Ku-C in- applied to Gaussian noise (Flandrin et al., 2004). The tech-
strument and is used as the reference mission for cross- nique was first adapted to process satellite altimeter SWH
calibration with other altimeters to provide consistent prod- measurements (Quilfen et al., 2018; Quilfen and Chapron,
ucts in the CMEMS framework. The Satellite with ARgos 2019; Dodet et al., 2020), and the algorithm is described in
and ALtiKa (SARAL) mission carries the AltiKa altime- detail in Quilfen and Chapron (2021). For the processing of
ter, which makes measurements at higher effective resolution the SLA data analyzed in this study, only limited modifica-
due to a smaller footprint obtained in the Ka band (8 km di- tions were made, and the algorithm is only briefly described
ameter vs 20 km on Jason-3) and a higher pulse repetition below.
rate. The altimeter on board Sentinel-3 is a dual-frequency Three main elements characterize the properties of the de-
Ku-C altimeter that differs from conventional pulse-limited noising algorithm: (1) the EMD algorithm that adaptively
altimeters in that it operates in delay Doppler mode, also splits the SLA signal on an orthogonal basis without having
known as synthetic aperture radar mode (SARM). SARM is to conform to a particular mathematical framework; (2) the
the primary mode of operation, providing ∼ 300 m resolution denoising algorithm that relies on high-frequency local noise
along the track. The SARM reduces instrumental white noise recovery and analysis; (3) an ensemble-average approach to
and is free of “bump” artifacts, which are caused by sur- estimate a robust denoised SLA signal and its associated un-
face backscatter variabilities, blooms, and rain-induced in- certainty.
homogeneities in the low-resolution mode (LRM) footprint,
adding spatially coherent error to the white noise (Dibar- 3.1 The EMD algorithm
boure et al., 2014). Still, current SARM measurements are
affected by colored noise, which is likely attributed to the ef- EMD is a data-driven method, often used as an alternative
fects of swell on SARM observations (Moreau et al., 2018; to wavelets in denoising a wide variety of signals. EMD de-
Rieu et al., 2021). For the CMEMS data set version avail- composes a 1D signal into a set of amplitude- and frequency-
able at the time of this study, the retracking used to process modulated components, called intrinsic modulation func-
the data is MLE-4 for Jason-3 and AltiKa and SAMOSA tions (IMFs), which satisfy the conditions of having zero
for Sentinel-3 (Taburet et al., 2021). Each altimeter makes mean and a number of extrema equal to (or different by
measurements at nadir along the satellite track, and the stan- one than) the number of zero crossings. IMFs are obtained
dard CMEMS processing provides data at 1 Hz with a ground through an iterative algorithm, called sifting, which extracts
sampling that varies slightly from 6 to 7 km depending on the high-frequency component by iteratively computing the
the altimeter. At this ground sampling, the average noise af- average envelope from the extrema points of the input sig-
fecting the range measurements is different for each altime- nal. The sifting algorithm is first applied to the input SLA
ter, with SARAL and Sentinel-3 showing significantly lower signal to derive the first IMF, IMF1, which is removed from
level of noise than Jason-3 (Taburet et al., 2021). The data the SLA signal to obtain a new signal on which the process
set available at CMEMS for the current analysis covers the is repeated until it converges when the last calculated IMF
period until June 2020 with a beginning in March 2013, June no longer has a sufficient number of extrema. The original
2016, and May 2016 for AltiKa, Sentinel-3, and Jason-3, re- signal is exactly reconstructed by adding all the IMFs. Fig-
spectively. ure 1 shows two sets of IMF for two passages of SARAL
Although only the quality-controlled CMEMS data are over the Gulf Stream area. Panels (a) and (g) show the two
used as input in our analysis, ancillary data are useful in SLA signals and the associated SWH signals for reference
supporting the analysis of SLA data. Indeed, since some of (red curves), and the other panels display the full set of IMF
the larger non-Gaussian SLA errors, correlated with high sea (six and four derived IMFs for these two cases, a number that
state conditions and rain or slick events, are expected to re- can vary with signal length and observed wavenumber spec-
main after the EMD analysis, SWH and radar cross-section trum). Shown in panels (b) and (h), IMF1 genuinely maps
(sigma-0) are also provided in the denoised SLA products the high-frequency noise in term of amplitude and phase,
to allow for further data analysis and editing. These are which can provide a direct approach to help remove high-
provided by the sea state Climate Change Initiative (CCI) frequency noise from the SLA signal. Local analysis of this
products, developed by the European Space Agency (ESA) high-frequency noise is used to predict and remove the lower-
and processed by the Institut Français de Recherche pour frequency noise embedded in other IMFs, as detailed below.
l’Exploitation de la Mer (IFREMER, Dodet et al., 2020). In panel (b), IMF1 is also associated with high-frequency
noise but shows non-stationary noise statistics that are re-
lated to changes in mean sea state conditions. As expected,
3 Methods the high-frequency noise of SLA increases with SWH. These
two examples are general cases, but IMF1 can also contain
The proposed denoising technique essentially builds on the geophysical information in cases where the SNR is locally
EMD technique (Huang et al., 1998; Wu and Huang, 2004; very high, for example in the presence of very large geophys-
https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 2022
1496 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals
ical gradients, or can show the signature of outliers related to A detailed description of the entire denoising scheme can
the so-called spectral hump (rain, slicks, etc.). Indeed, de- be found in Quilfen and Chapron (2021), and the main steps
pending on the SNR and in specific configurations for the are given in Appendix A.
numerical sifting algorithm, the type of large SLA gradient Figure 2 provides illustrations of the general approach
signature shown in IMF2 (panel i) can very well show up taken to denoising SLA signals. They show the PSD of SLA
in IMF1, in case there are no detectable extrema between (black curves), the associated IMFs (blue curves), and the
measurements number 2 and 8. Since IMF1 analysis is at IMFs of a white noise (red curves) whose standard devia-
the heart of the denoising strategy described below, careful tion has been adjusted to fit the SLA background noise be-
preprocessing of IMF1 is necessary before denoising the full tween 30 and 15 km wavelength. It is presented for the Ag-
signal. ulhas Current area, panel (a), and for the Gulf Stream area,
panel (b). For clarity, only the first three IMFs are shown.
3.2 The denoising scheme As expected, for white noise, the EMD filter bank is com-
posed of a high-pass filter with the IMF1, and a low-pass
Flandrin at al. (2004) applied EMD to a Gaussian noise sig- filter bank with the higher ranked IMFs. A similar structure
nal to demonstrate that the IMF1 has the characteristics of is observed for the IMFs of the SLA signal with identical
a high-pass filter while the higher-order modes behave sim- cut-off wavelengths, which is the result of the noise shap-
ilarly to a dyadic low-pass filter bank, for which, as they ing the frequency content of the SLA signal. This similar-
move down the frequency scale, successive frequency bands ity shows the consistency of separate denoising of each IMF
have half the width of their predecessors. Unlike Fourier or of rank n > 1 using the estimated noise variance given by
wavelet decompositions for which the noise variance is in- Eq. (2). The IMF1 PSDs of SLA and white noise have a sim-
dependent of the scale, the noise contained in each IMF is ilar shape, both containing mostly high-frequency noise, but
now “colored” with a different energy level for each mode. with higher SLA PSD values at scales > ∼ 20 km, which is a
Flandrin et al. (2004) deduced that the variance of the Gaus- consequence of the large modulation of the SLA noise by the
sian noise projected onto the IMF basis can be modeled as varying sea state conditions and the inclusion of geophysical
follows for the low-pass filter bank: information such as that related to very large SLA gradients
showing high SNR. These higher PSD values are even more
var(hn (t)) ∼ 2(α−1)n , (1) important in the Gulf Stream area due to larger variety of sea
states encountered and sampled by the altimeter passages.
where hn (t) are the IMFs of rank n > 1 and α depends on the A is an important factor to adjust because it is directly re-
Hurst exponent H of the fractional Gaussian noise. For the lated to the improvement obtained in the SNR. Kopsinis and
altimeter data set, the white noise assumption is made, fol- McLaughin (2009) perform the optimization of the A factor
lowing studies showing a quasi-white noise spectrum (Zaron by simulating a variety of input signals and SNR values. Fol-
and De Carvalho, 2016; Xu and Fu, 2012; Sandwell and lowing these results, first approximate values were tested for
Smith, 2005). It corresponds to H = 0.5 and α = 0 in Eq. (1). the altimeter data set, with a careful analysis of the obtained
Flandrin et al. (2004) then numerically derive, using denoised SLA measurements. However, the adjustment of A
Eq. (1) and for different values of H , the relationship be- for the different altimeters was refined. Indeed, the values of
tween the IMF’s variance En , for n > 1, and the variance of A showing the best SNR improvement on average may de-
IMF1, E1 . For a white noise, this gives pend on the average SNR of the input signal, which is differ-
ent for each altimeter. Section 4.2 details the practical rule for
E1
En = 2.01−n . (2) determining A that uses an approach to make the denoised
0.719 SLA PSD consistent with the mean PSD of the unfiltered
With the EMD basis, the noise energy decreases rapidly with data minus the PSD of the white Gaussian noise (WGN) es-
increasing IMF rank, ∼ 59 %, 20.5 %, 10.3 %, 5.2 %, 2.6 %, timated in the range of 15–30 km, and consistent for different
of the total energy for the top five IMFs, respectively. The altimeters.
first four IMFs account for ∼ 95 % of the noise energy.
Equation (2) therefore gives the expected noise energy in 4 Results
each IMF to determine the different thresholds below which
signal fluctuations can be associated with noise. The thresh- 4.1 Examples
old formulation introduces the constant factor A, which is a
control parameter that can be adjusted for different altimeters The two cases (Fig. 1) show AltiKa SLA measurements in
depending on their characteristic noise levels: the Gulf Stream area, and Fig. 3 illustrates the EMD denois-
p ing principles for these examples. Described in Sect. 3.2,
Tn = A En , (3) denoising a segment of SLA data after an initial expansion
into an IMF set is a two-step process: (1) wavelet analysis of
with n being the rank of the thresholded IMF. IMF1 to separate and evaluate the high-frequency part of the
Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022
Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1497 Figure 1. SARAL SLAs (a and g) in the Gulf Stream area for cycle 106 and passes 53 (a–f) and 597 (g–k), and the corresponding IMFs obtained from the EMD decomposition (b–f and h–k). SWH associated with the SLA is shown in red curves (a and g). All units in meters. Gaussian noise and any geophysical information embedded threshold (i.e., Eq. 2), which maps the large SLA gradient in IMF1; (2) EMD denoising of a set of 20 realizations of in the Gulf Stream and mesoscale features near 70 km wave- reconstructed noisy SLA series to estimate a mean denoised length with some eddies appearing well above the threshold SLA series and its uncertainty. and other smaller amplitude oscillations that will be canceled Pass 597, panels (g) to (k), is associated with rather low in the second denoising step. As shown in Fig. 2, the SNR in- sea state conditions with little variability, and IMF1 (black creases rapidly for IMF2 compared to IMF1. In this case, the curve, panel h) has little amplitude modulation, but rather uncertainty attached to the denoised SLA is almost constant large phase modulation due to the the high SNR in sev- below 1 cm, as shown in Fig. 3h. eral portions of the segment (minor alternation of minima AltiKa pass 53 crosses the Gulf Stream 19 d before, but and maxima). Because of this relatively large phase modu- this is a very different situation. Quite frequently, such a case lation, a significant portion of the IMF1 is identified in the corresponds to high and variable sea state conditions with first step as “useful signal” by the wavelet analysis. In the abrupt changes in SWH, as shown in Fig. 1. Strong westerly second step however, this residual IMF1 signal will be al- continental winds were present for several days before the most completely removed. Indeed, it is well below the SNR AltiKa passage, which turned to the northwest the day be- prescribed by using the high-frequency noise jointly derived fore. SWH was less than 2 m near the coast between records from the IMF1 wavelet processing and the threshold values 80 and 120, then a first large increase occurred on the north- set with Eqs. (A1), (2), and (3) (blue lines in Fig. 3). Only ern side of the Gulf Stream near record 60, and a second a small modulation between data records 80 and 90 there- on its southern side to reach sea state conditions with SWH fore shows up in the denoised SLA signal. Figure 3i shows > 5 m. Unlike the first case, the IMF1 thus shows a large IMF2, and its associated threshold derived from the IMF1 modulation in amplitude, and relatively small modulation https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 2022
1498 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals
to be associated with larger noise values correlated with the
high SWH values. It will remain an outlier, however, and will
be adequately associated with the largest uncertainty in this
data segment. Indeed, the uncertainty calculated as the stan-
dard deviation of the set of denoised signals increases with
sea state, doubling along this data segment. Overall, these
two examples confirm that the proposed denoising process
adapts well to varying sea state conditions.
In cases where sigma-0 blooms or rain events corrupt lim-
ited portions of a data segment, and for which the data editing
step was not performed, the impact is more difficult to ana-
lyze. It will depend on the magnitude and length of the asso-
ciated errors which can vary greatly. However, the proposed
EMD denoising process is not a data editing process and the
results are certainly still affected by some of the largest er-
rors. It should benefit from improvements in data editing pro-
cedures and retracking algorithms that will be used for future
CMEMS products.
4.2 EMD denoising calibration by PSD adjustment in the
30–100 km wavelength band
For SLA measurements performed by a given altimeter in-
strument, the mean SNR is expected to vary primarily with
sea state. The mean SNR is then a function of the climatolog-
ical distribution of sea state conditions that are dependent on
ocean basins and seasons. The proposed denoising approach
can efficiently adapt to the local SNR, allowing for a single
global value for the control constant A in Eq. (3). However,
since the noise statistics vary greatly with the average sea
state conditions, it is useful to show how such variability can
impact the results when a single value of A is used in global
Figure 2. Mean PSD of the first three IMFs (first = solid; sec- SLA processing. A two-step sensitivity study is performed
ond = dashed; third = dotted) for white noise (red curves) and below, which first determines specific A values for different
SARAL SLA along-track measurements (blue curves), and mean regions, and then shows how the use of an overall A value
PSD of the corresponding noisy (thick black line) and denoised
impacts the results. A few areas are defined corresponding
(thin black line) SLA measurements. The PSD is the average of
to different climatological sea state conditions, namely the
PSDs computed over all data segments covering the years 2016–
2018, for the Agulhas (10–35◦ W; 33–45◦ S, a) and Gulf Stream Gulf Stream region, the Agulhas Current region, an area in
(72–60◦ W; 44–32◦ N, b) regions. The green line is for the PSD the southern Indian Ocean, and an area in the central Pacific
of the SLA high-frequency noise estimated from the SLA’s IMF1 Ocean. The precise coordinates of the regions are given in the
(solid blue line). legend of Fig. 4. The analysis was then performed using Al-
tiKa measurements, as the AltiKa PSD curve shows a well-
defined, expected white noise plateau in the high-frequency
range, 15–25 km, as shown in Fig. 4, which is not the case for
in phase. Since the phase modulation of IMF1 is primarily Jason-3 and Sentinel-3. We see that the height of the noise
high-frequency for wavelet analysis, it is analyzed as noise, plateau depends on the climatological sea state conditions,
and the useful signal is almost zero everywhere except for with the highest value for the southern Indian Ocean, fol-
the largest IMF1 values. Furthermore, because the threshold lowed by the Agulhas, the Gulf Stream and the central Pacific
value computed from the estimated noise is also larger (than Ocean. This is the effect of the dependence of the instrumen-
for the first case), the IMF1 useful signal is completely re- tal noise, after retracking, on SWH since the hump artifact
moved in the second EMD denoising step. This highlights does not depend on wave height (Dibarboure et al., 2014).
the potential of the denoising approach to handle variable sea The two-step analysis for each region then follows.
state conditions. Nevertheless, the IMF2 processing (Fig. 3c)
shows that a modulation of the SLA is found to be significant 1. For each region, a set of discrete A values within a range
near the beginning of the IMF2 record, which may appear corresponding to that found in Kopsinis and McLaughin
Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022
Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1499
Figure 3. SARAL short data segments in the Gulf Stream area for cycle 106 and passes 53 (a–d) and 597 (e–h). Panels (a) and (e): noisy
(black dotted) and denoised (red) SLA; panels (b) and (f): IMF1 (black), useful signal (red) retrieved from wavelet denoising of IMF1, and
thresholds (blue) to be applied in IMF1 EMD denoising; panels (c)) and (g): IMF2 (black) and thresholds (blue) to be applied in IMF2 EMD
denoising; panels (d) and (h): noise (black) retrieved from IMF1 wavelet denoising and uncertainty (red) attached to the denoised SLA. All
units in meters on the y axis; x axis data are the record numbers.
(2009) are used to process the 3-year data set and obtain a power law in k 0 . This artifact is caused by backscat-
an average PSD of denoised measurements for each dis- ter inhomogeneities in the altimeter footprint associated
crete A value. An optimal A value, for each region, is with sigma-0 blooms and rain cells (Dibarboure et al.,
then found as the one that gives the best fit between the 2014), and it indeed contaminates altimeter measure-
observed mean PSD and a mean SLA PSD calculated ments much more frequently in tropical regions. Con-
as the sum, over the data segments covering 3 years, of flicting results are discussed in Dibarboure et al. (2014)
the denoised SLAs and a WGN whose average standard regarding the spectral shape of the hump artifact, show-
deviation is calculated to fit the mean observed PSD in ing that it can be distributed as white noise or as a dome-
the 15–25 km range. The best fit between PSDs is esti- shaped figure depending on the analysis approach. In
mated as the root mean square deviation (RMSD) in the practice, it can also depend on the data editing, on the
30–100 km range. A minimum value for the RMSD is waveforms retracking, and on the way the PSDs are
then found in the prescribed range of A. The denoised computed. Dibarboure et al. (2014) showed that a flat
SLA PSD corresponding to this optimal value of A is hump PSD is found when a large amount of long data
the theoretical PSD that gives the best fit with the ob- segments are used to compute the PSD, which is not
served PSD for the white noise observed in the 15– the case in the tropical oceans, where hump artifacts are
25 km range, and is referred to as the “best fit” PSD frequently edited, and then reduce the length of continu-
in Fig. 4. The “best fit plus WGN” PSD is in excel- ous data segments. The PSD resulting from the classical
lent agreement with the observed PSD in three of the analysis (Fu, 1983; Le Traon et al., 2008; Dufau et al.,
regions. In the central equatorial Pacific, the observed 2016) performed to estimate the SLA spectral slopes is
difference may be related to the so-called hump artifact also shown in Fig. 4 (referred to as “observed minus
that can cause a deviation of the total noise PSD from WGN” PSD). It is in good agreement down to 50 km
https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 2022
1500 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals
with the “fitted PSD” for the three regions for which the lated using the median absolute deviation from zero of
best fit plus WGN PSD also agrees well with the ob- IMF1. For this reason, processing IMF1 using wavelet
served PSD. The differences below 50 km wavelength analysis is an important step to separate, as much as
may be due to the fact that the hump artifact resulting possible, the possible useful geophysical signal in IMF1
from contamination of portions of a number of data seg- from outliers, and to estimate the underlying Gaussian
ments is not well distributed as white noise for our data noise.
segment collection.
The reasoning used above to set the control parameter A is
2. An optimal value of A is therefore obtained for each based on the assumption that the measurements are affected
region, ranging from 1.8 to 2.2. To show the results ob- by additive Gaussian noise with a known wavenumber de-
tained when the same value of A is used for all regions, pendence (in this case in k 0 , as in studies correcting mean ob-
the same value of 1.8 was used to EMD-process sim- served PSDs from mean white noise), whereas this may not
ulated data calculated as the sum of the denoised SLA be the case in many along-track data segments where artifacts
data corresponding to the theoretical PSDs of step 1 for related to sigma-0 blooms, rain events, or high sea state con-
each region (best fit PSD in Fig. 4) and a white noise es- ditions may produce deviations of the noise from a Gaussian
timated in the 15–25 km range. The PSDs obtained by process (as shown in Fig. 1b). Depending on the region, the
processing these simulated data with this single value “observed PSDs” shown in Fig. 4 may be strongly shaped by
of A are called “retrieved” PSDs in Fig. 4 and are very these events, and this may explain the observed differences
close to the best fit PSDs in all regions. This indicates between the best fit and “observed – WGN” PSDs, shown in
that the process as a whole can provide a set of denoised thin black and green curves, respectively. To further verify
measurements with a realistic energy distribution in the the consistency of the EMD denoising and to show the role
observed wavelength range. Further, any variation in the of IMF1 processing, the 3-year data set of denoised SLAs
prescribed thresholds, which depend on the setting of A, in the Gulf Stream region (PSD shown in Fig. 4a) was con-
is accounted for in the uncertainty parameter attached sidered the true geophysical signal of the SLAs, thus pre-
to the denoised measurements. Another way to assess scribing a theoretical PSD. For each data segment of these
the choice of the setting of parameter A is to perform “theoretical” SLAs, a white noise of 1.8 cm standard devia-
a sensitivity study using Monte-Carlo simulations. One tion was added to provide a simulated noisy SLA segment.
thousand WGN series and associated IMFs were gener- This data set was then processed with the EMD denoising al-
ated. The IMF1s contain the high-frequency component gorithm. Figure 5 shows that the theoretical, EMD denoised,
of the noise series, whose spectrum is represented by and “Noisy – WGN” PSDs are in excellent agreement over
the red solid line in Fig. 2. For each of the 1000 IMF1 the entire wavelength range, in contrast to what is obtained
series, the threshold for the EMD denoising process was with the observed SLA shown in Fig. 4. The same coherence
computed using Eqs. (A1), (2), and (3), and applied to is obtained regardless of the region analyzed. This means that
test the IMF1s against the expected noise. The results the EMD denoising process is fully consistent in the case of
show that more than 98.5 %, 99 %, and 99.5 % of the measurements contaminated by a Gaussian noise, which also
IMF1 data values are below the threshold with A val- suggests that the hump artifact may deviate more or less from
ues of 1.8, 2, and 2.2, respectively, which is the range a white noise distribution for our data set.
of best fit values found in step 1 above for the different In this simulation, the SNR is close to 1 on average near
regions. EMD denoising is therefore effective in cases 50 km wavelength, as shown in Fig. 5, but can be greater than
for which the additive noise is close to the Gaussian, 1 locally in a wavelength range down to 30 km. Such small
and is not very sensitive to variations found in A for mesoscale geophysical information emerging from the noise
the different regions. IMF1 values above the prescribed level can be retrieved from IMF1 using dedicated wavelet
threshold can therefore be associated with geophysical denoising analysis. As shown in Fig. 5, the average PSD of
information with good statistical confidence, especially IMF1 shows the plateau of high-frequency noise, but also
since their significance will be tested further since de- significant energy content over a wider wavelength range as-
noising is achieved with an ensemble average of noisy sociated with both lower-frequency noise and geophysical
processes. To process AltiKa globally, A is set to 1.925, information. The PSD curves of the IMF1 and the simu-
which is the exact average value found for the four re- lated SLA intersect between 50 and 30 km wavelength. After
gions analyzed. wavelet decomposition of the IMF1, the wavelet denoising
The problem to consider further is the presence, in lim- scheme specifies the maximum level to be retained for geo-
ited portions of the processed data segments, of out- physical signal recovery. In the general case, only the level
liers associated with high waves or artifacts caused by containing the finest scales is systematically discarded, and
sigma-0 blooms or rain events. These will likely appear Fig. 5 shows the PSD of the signal recovered from IMF1 af-
in IMF1 and IMF2 series and the most energetic events ter using the Huang and Cressie (2000) denoising scheme
will not be thresholded since the thresholds are calcu- (red dashed curve). The wavelet denoising acts as a low-
Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1501
Figure 4. Mean PSD of SARAL SLA along-track measurements: observed (thick black), best fit (thin black), best fit plus WGN (red),
retrieved (dashed red), and observed minus WGN (green). WGN is estimated as the average in the range of 15–25 km of the mean observed
PSD (bold black curve). The PSD is the average of PSDs computed over all data segments covering the years 2016–2018: the Gulf Stream
(72–60◦ W, 44–32◦ N, a), the South Indian (80–110◦ E, 60–40◦ S, b), the Agulhas (10–35◦ E, 45–33◦ S, c), the Central Pacific (170–150◦ W,
10◦ S–10◦ N, d) regions.
pass filter with a sharp cut-off near 25 km wavelength and events, there is no simple relation between the noise associ-
a significant amount of noise is also filtered out at longer ated with the hump artifact and sigma-0 that would allow for
wavelengths. In this simulation, the processing results in the such a practical rule. EMD denoising of real SLA measure-
recovery (red curve) of the full PSD (thin black curve) of ments is therefore likely to still be contaminated by outliers
the simulated signal because the A parameter has been set in limited portions of a number of along-track data segments,
to do so (A = 1.65), although in practice, and even though and future improvements will depend on better data editing
the SNR was dramatically improved (more than 97 % of the and implementation of the latest retracking algorithms (Pas-
IMF1 noise canceled), the geophysical signals with the low- saro et al., 2014; Thibaut et al., 2017; Moreau et al., 2021).
est SNR were also filtered out. In this sense, and as expected,
the SLA signal for the real data will only be partially resolved
in the small mesoscale range. In specific cases, further filter- 4.3 Building a multi-sensor data set with consistent PSD
ing can be applied by setting a different maximum level for
The EMD denoising algorithm is then found to be robust and
wavelet denoising of the real data, as outliers can contribute
consistent in processing the AltiKa measurements. A work-
strongly to the IMF1 in the wavelength range of 10–50 km.
able rule can be defined to adjust the method to provide a
For example, a practical rule can be considered by further
global data set of denoised SLA measurements whose PSD
constraining IMF1 denoising when more than a given per-
are regionally consistent with the expected SLA geophysi-
centage of a processed data segment is associated with high
cal signals. Such an approach is not easily applicable or nu-
seas. For outliers associated with sigma-0 blooms and rain
merically consistent for the Sentinel-3 and Jason-3 measure-
https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 20221502 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals
noise level and shape above 25 km wavelength. The same
value of A was therefore used to set the noise thresholds for
Sentinel-3, and it yields noise-free measurements PSDs in
near perfect agreement with AltiKa. Note that, unlike AltiKa,
the Sentinel-3 measurements are not sensitive to the hump ar-
tifact, thanks to the SAR processing, and this difference is not
apparent in the overall results shown in Fig. 6. Although the
PSD shape of the Sentinel-3 noise has often been referred to
as red noise, there are no published results showing that this
is the case in the range of interest, i.e., wavelengths > 30 km.
The similarity to the AltiKa PSD in this range might indi-
cate that this is not the case, and this justifies the choice
of retaining the white noise configuration for the Sentinel-
3 EMD processing. However, the processing is capable of
dealing with other Gaussian noise figures, and Fig. 6a shows,
for information, the result obtained for Gaussian noise with
a Hurst exponent of 1, corresponding to a PSD in k −1 rep-
resented by a green solid line in the 15–30 km wavelength
Figure 5. Mean PSD of SARAL SLA along-track measurements: range. The PSD associated with the denoised measurements
simulated noise-free SLAs (thin black), simulated noise-free SLAs is as expected slightly lower than the white noise case in the
+ simulated WGN with 1.8 cm SD (thick black), EMD-denoised range of 30–100 km.
SLAs (solid red), IMF1 of SLAs (dotted red), and noise-free por- For Jason-3 and the 3-year data set analyzed, the control
tion of IMF1 obtained after IMF1 wavelet processing (dashed red).
constant A was adjusted and set to 2.4 in order to obtain the
The above PSDs are computed as the average of PSDs obtained for
best fit of the PSDs of denoised measurements with AltiKa.
all individual data segments covering the years 2016–2018, and the
Gulf Stream region (72–60◦ W; 44–32◦ N). The PSD shown as a Fig. 6 thus shows a high degree of consistency between the
green line is obtained by subtracting from the thick black line the PSDs of the three altimeters, although each one only par-
mean WGN (std = 1.8 cm) shown as the black dashed line (averaged tially resolves the true geophysical content in the 30–100 km
over 15–30 km). range, with Jason-3 doing worse than the other two. Indeed,
the similarity of Jason-3 denoised PSD with the other two,
while the noise level is significantly higher, indicates that
ments. Their PSDs do not exhibit the expected white noise the SNR has been less improved. However, this difference
plateau in the 10–25 km wavelength range, Fig. 6. The red- will be taken into account in the uncertainty parameter at-
type noise in Sentinel-3 measurements has already been dis- tached locally to the denoised measurements, as documented
cussed and analyzed in several studies, and has been shown in the next section. Note also that, although the EMD pro-
to be mainly related to the effects of swell on SARM obser- cess adaptively accounts for the noisier Jason-3 measure-
vations (Moreau et al., 2018; Rieu et al., 2021). The reason ments, a higher A threshold is necessary to obtain the best
why the Jason-3 PSD also shows a tilted PSD in the high- fit with the other altimeter PSDs. This results from the need
frequency range is more puzzling. One possible explanation to compensate for the poor data editing (especially rain flag)
is that it results from poor data editing (especially the rain of the Jason-3 measurements, as discussed at the beginning
flag) for Jason-3. Indeed, while an effective rain flag was of the section. Indeed, since the outliers affect only limited
used for AltiKa so that rain has little influence on data quality portions of an analyzed data segment, they do not affect the
(Verron et al., 2021), this is not the case for Jason-3, which estimation of the EMD denoising thresholds, which are cal-
is therefore likely to be more impacted by rain events. As- culated using the median absolute deviation from zero of the
sociated errors may shape the noise distribution differently IMF1 noise. The thresholds are more tuned to the instrumen-
than white noise, as discussed in the previous section. Indeed, tal noise rather than to the total noise, and outliers of large
we found many more short segments of continuous measure- amplitude are not removed. They have a significant impact
ments in the AltiKa data set than in the Jason-3 data set, both on the denoised PSD, so a larger A is needed to achieve sim-
of which are distributed in the same CMEMS product, due ilar PSD levels for Jason-3.
to more efficient data editing. Therefore, the adjustment of For reference, Fig. 6 shows a k −4 law that indicates steeper
the EMD denoising process for Jason-3 and Sentinel-3 was slopes for the Gulf Stream and Agulhas regions, a slope close
performed by using the AltiKa results as reference. to k −4 in the south Indian region, and a flatter slope in the
For Sentinel-3, Fig. 6 shows that its PSD for all analyzed central Pacific. The variance in the 30–100 km range is the
regions is in excellent agreement with AltiKa’s PSD over the highest in the Gulf Stream region, followed by the Agulhas,
entire wavelength range down to 25 km, which is a striking south Indian, and central Pacific regions, in descending order,
result, showing that the two altimeters have similar average in agreement with the results of Chen and Qiu (2021).
Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1503
4.4 Spectral slopes seasonality agreement with the results of Rocha et al. (2016), who used
acoustic Doppler current profile (ADCP) measurements in
Seasonal variations in SSH in the small mesoscale range, the Drake Passage, and disagrees with Vergara et al. (2019)
with a wavelength less than about 100 km, are very diffi- and Chen and Qiu (2021), who used the standard approach
cult to analyze because the noisy SLA spectral slopes are applied to altimeter data. As mentioned, high sea conditions
strongly shaped by errors related to the hump artifact, not make it difficult to assess the results obtained by the differ-
dependent on the sea state, and by the instrumental and pro- ent approaches, and better data editing and retracking algo-
cessing noise which is correlated with sea state conditions. rithms (Passaro et al., 2014; Thibaut et al., 2017; Moreau
Although the behavior of the resulting total noise is not well et al., 2021) would improve the current analysis. The Agul-
understood, it is genuinely postulated that the total SLA noise has region, number 5, which experiences mixed sea condi-
in the 1 Hz measurements can be considered to be WGN tions, shows no seasonality, which is in agreement with the
when a large amount of long segments is used to calculate results of Chen and Qiu (2021). Three regions, 1, 4, and 6,
the spectrum. This enabled the empirical study of seasonal show seasonality, insensitive to the filtering of high sea con-
variations in SSH by removing an average noise PSD from ditions. Strong seasonality in mesoscale dynamics on scales
the average PSDs of altimeter measurements (e.g., Vergara of 1–100 km, driven by turbulent scale interactions, are found
et al., 2019; Chen and Qiu, 2021). However, the applicability in the Gulf Stream area using numerical modeling experi-
of this assumption may not be verified at the regional level, ments and in situ observations (Mensa et al., 2013; Callies
as suggested by the results presented in Figs. 4 and 5. This et al., 2015), confirming the present altimetry results. Chen
certainly also depends on the performance of the data edit- and Qiu (2021) also show this strong seasonality in the Gulf
ing. Therefore, the adoption of an alternative approach based Stream region and in region 6, west of Australia, as also ob-
on the analysis of the along-track EMD-denoised SLA mea- tained in our results. Conversely, we find seasonality in the
surements, rather than on the denoising of the SLA spec- western tropical Atlantic, region 4, not reported by the Chen
trum, is likely more suited. It has also been shown that sea and Qiu (2021) study. Overall, for regions showing season-
state-related errors are essentially removed by the EMD pro- ality in the small 30–100 km mesoscale range that is appar-
cessing. Figure 6 shows that consistent spectral slopes of the ently unaffected by high sea states events, SLA variability is
denoised SLA are well obtained for the different altimeters. found to be greater in winter of each hemisphere, consistent
Hereafter, we only use AltiKa denoised measurements be- with stronger atmospheric being a source of enhanced sub-
cause the period covered is much longer for more consistent mesoscale ocean dynamics (Mensa et al., 2013). For refer-
analysis of seasonal variations. The data used cover 6 years, ence and evaluation of the data, and although different dy-
from summer 2013 to winter 2018–2019, and the eight dif- namics may be at work, a k −4 spectral slope is shown in
ferent regions shown in Fig. 7 are defined to cover various Fig. 8 for the 30–120 km wavelength range. It shows that, in
climatological sea state conditions and expected energy lev- the small mesoscale range, the steepest slope calculated from
els in the small mesoscale variability of SLA. the PSD of denoised SLA measurements is found in the Gulf
For each region, the average SLA spectrum is shown in Stream region with a slope between k −4 and k −5 . It is close to
Fig. 8 for the boreal summer and winter, for the entire data k −4 in the Agulhas region and the high seas 7 and 8 regions.
set, and for a data set limited to segments having more In other regions related to the intra-tropics, flatter slopes are
than 80 % of measurements with SWH < 4.5 m. This arbi- found, consistent with increased energy from internal tides
trary SWH threshold corresponds to the 90th percentile of and gravity waves (e.g., Garrett and Munk, 1972; Tchibilou
the global data set and is intended to limit the influence of et al., 2018). Overall, these different results are found to be
possible remaining outliers associated with extreme sea state consistent with studies using modeling experiments or obser-
conditions. In regions of high climatological sea state, this vations from altimetry and in situ data.
thresholding will significantly limit the available data seg-
ments used to calculate the spectrum. 4.5 Uncertainties in denoised sea level anomalies
Distinct regions can be considered in Fig. 8. The intra-
tropical regions, 2 and 3, show no seasonality, a result in For a processed data segment, the resulting denoised SLA
agreement with previous studies (Vergara et al., 2019; Chen segment is the average of 20 realizations of the denoising
and Qiu, 2021). In these regions, stable low sea state condi- process. An uncertainty ε, which characterizes the expected
tions cannot introduce strong errors in the analyses. In the error attached to each denoised SLA in the data segment,
rough southern oceans, regions 7 and 8 (the Drake Passage) is then calculated as the local standard deviation of the set
show a small apparent increase in small mesoscale energy of denoised SLA profiles corresponding to 20 random real-
in the austral winter, disappearing when the high sea state izations of the noisy SLAs profiles. Since the noise process
threshold is applied. In the latter case and for the Drake Pas- used to generate the set of random realizations is the high-
sage, 130 and 73 AltiKa passes satisfy the criterion and were frequency part of the observed Gaussian noise derived from
used to estimate the mean spectrum for the boreal JJA and IMF1, ε readily accounts for the various errors affecting the
DJF, respectively. This suggests the absence of seasonality, in data (i.e., instrumental and processing noise and remaining
https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 20221504 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals Figure 6. Mean PSD of observed and denoised SLA along-track measurements for Jason-3 (blue), AltiKa (black), Sentinel-3 (red), and four regions: Gulf Stream (panel a; 72–60◦ W, 44–32◦ N), Agulhas (panel c; 10–35◦ E, 45–33◦ S), south Indian (panel b; 80–110◦ E, 60–40◦ S), and central Pacific (panel d; 170–150◦ W, 10◦ S–10◦ N). The PSD is the average of PSDs computed over all data segments covering the years 2016–2018. For illustration, the green curve in panel (a) shows the PSD of denoised Sentinel-3 SLA when processed with EMD and the hypothesis of a Gaussian noise following a k −1 slope (pink noise, Hurst exponent = 1, shown as a green solid line in the range of 15–30 km). The dark solid line shows a k −4 slope in the range of 30–150 km. Figure 7. Yearly averaged SWH (m) computed over 2016–2018 from the Climate Change Initiative L4 products. Dashed black boxes define the eight areas analyzed in the section. Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022
Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals 1505 Figure 8. Mean PSD (m2 per cycles per kilometer) of AltiKa denoised SLA in boreal summer (solid black curve, JJA) and in boreal winter (dashed black curve, DJF). The red curves show the results using a data set limited to segments showing more than 80 % of data with SWH < 4.5 m. The numbered eight panels correspond to the eight areas shown in Fig. 7. For reference, a PSD law in k −4 is shown in the wavelength range of 30–120 km, green line. outliers), but it also represents the uncertainty related to the data editing performed for each instrument. They are how- local SNR itself via the IMFs thresholding. The larger the lo- ever in agreement with other previous studies (e.g., Dufau et cal SNR, the higher the SLA modulation above the expected al., 2016; Vergara et al., 2019). noise threshold and the lower the standard deviation ε, and The uncertainty parameter ε thus characterizes the error at- vice versa. tached locally to each denoised SLA measurement, account- The probability density function (PDF) of IMF1, of the ing for the variations in the SNR but also reflecting the errors high-frequency noise, and of ε is shown in Fig. 9. Sentinel- and choices made in the denoising process. Thus, the choice 3 displays the best statistics, which is mainly a result of of imposing similarity on the Jason-3, AltiKa, and Sentinel- reduced errors related to the hump artifact in the SARM 3 PDFs, although the SNR of Jason-3 is significantly lower processing. Jason-3 has significantly higher levels of noise. on average, implies less improvement in the SNR of Jason-3, These results are certainly strongly impacted by the different resulting in worse statistics for ε. https://doi.org/10.5194/essd-14-1493-2022 Earth Syst. Sci. Data, 14, 1493–1512, 2022
1506 Y. Quilfen et al.: Towards improved analysis of short mesoscale sea level signals
Figure 9. Probability density function (PDF, %) of the absolute values of IMF1 (m, a, solid lines), IMF1 HF noise (m, a, dashed lines), and
ε (m, b), for Jason-3 (blue), AltiKa (black), and Sentinel-3 (red).
The spatial distribution of ε is shown in Fig. 10. It is ucts. It relies on (1) the EMD algorithm that adaptively splits
mainly characterized by higher values in high-sea regions, in the SLA signal into a set of empirical functions that share
regions with high probability of rain events such as intertrop- the same basic properties such as wavelets, but without hav-
ical convergence zones, and also in regions for which the ing to conform to a particular mathematical framework; (2) a
SLA variance is larger in the 30–120 km wavelength range, denoising algorithm that relies on a thorough and robust anal-
associated with a lower SNR. Interestingly, similarities are ysis of the local Gaussian noise affecting the SLA data over
found with the altimeter 30–120 km SSH variance map an- the entire wavenumber range; (3) an ensemble-average ap-
alyzed by Chen and Qiu (2021). A more detailed analysis proach to estimate a robust denoised SLA signal and its as-
of the distribution of ε is beyond the scope of this study but sociated uncertainty; (4) a calibration of the method to pro-
certainly deserves further investigation. vide a realistic distribution of SLA variability by adjusting
the mean level of the PSD function.
It is therefore useful to compare our approach with the
5 Key elements of comparison with other
CMEMS products, but also with the Data Unification and
distributed products
Altimeter Combination System (DUACS) experimental 5 Hz
products distributed by the Aviso+ center, as the latter prod-
In this study, we highlight key features that make our ap-
ucts include, among several differences from CMEMS pro-
proach to denoising SLA data different and more attractive
cessing, high-frequency noise correction (Tran et al., 2019)
than the approaches currently used in other distributed prod-
Earth Syst. Sci. Data, 14, 1493–1512, 2022 https://doi.org/10.5194/essd-14-1493-2022You can also read
